An analytic continuation of (REF ) [1]} gives rise to a complete basis of positive
energy modes, i.e., the Kruskal modes [2]}. Then one can use
the Schwarzschild mode and the Kruskal mode to expand the Dirac field,
respectively, leading to the Bogoliubov transformations between
annihilation operator and creation operator under the Schwarzschild
and Kruskal coordinates [3]}. After properly normalizing the
state vector, the vacuum state and excited state of the Kruskal
particle in the single-mode approximation are given by
\(&|0\rangle =\cos \beta |0\rangle _{I}^{+}|0\rangle _{II}^{-}+\sin \beta |1\rangle _{I}^{+}|1\rangle _{II}^{-},\\&|1\rangle =|1\rangle _{I}^{+}|0\rangle _{II}^{-},\) 
